Time-Delayed Feedback Channel Design: Discrete Time H1 Approach

Marcella M. Gomez*, Seungil You* and Richard M. Murray

Abstract— This paper proposes a method of improving per-

formance of scalar discrete-time systems with substantial delay

by adding additional delayed feedback channels (i.e. imposing a

distributed delay feedback). The optimal weights for the added

feedback channels are found using optimization techniques. In

particular, we reduce the H1 norm of the closed loop transfer

function with multiple delayed feedback using techniques from

static output feedback design. We impose constraints on the

feedback gain in order to highlight the effectiveness of the

distribution. In this manner, improvement on performance is

a result of the distribution and not a change in the overall

effective gain. The concept of applying a multiple delayed

feedback channel is inspired by biological systems, where

substantial delays can be present in feedback control. To show

the effectiveness of this idea we apply our method to an example

of a scalar genetic autoregulatory network. The constraint on

the gain allows one to implement the feedback in a genetic

regulatory network without having to change the reaction rates.

A possible method of synthesizing such a system in a wet lab is

explained in more detail. Finally, stability results indicate the

possibility of stabilizing an unstable system with added delayed

feedbacks (by adding larger delays). This approach may also

be applicable to systems with large delays in which simple

controllers are needed due to limitations in computational

power. This paper motivates and provides preliminary results

towards direct design of purely delay based controllers for

network systems with large delays.

I. INTRODUCTION

Time delay in systems is an important uncertainty toconsider and should be treated carefully. Much work hasbeen done to deal with the existence of time delay in systems,from stability analysis to controller synthesis; see [1] and [2]for an excellent survey and references therein.

Although many tools exists for feedback controller design,such as PID, LQR, and H1 [3], implementing such con-trollers in a synthetic biological network is not trivial, dueto a lack of computational components in synthetic biology[4]. In addition, controller design for systems with inherentlylarge delays is a relatively new problem brought about bysynthetic biology and large network systems. Although, someprocesses in biology operate on fast time scales, some proteinregulated networks can have substantial delays. For example,protein folding can impart significant delays, depending onits size [5].

We look to existing biological systems for a possibleanswer. Despite having delays and inherent stochastic pro-cesses, biological systems remain robust in the presenceof noise. In particular, this paper is motivated by recent

*Authors made equal contributionsThe authors are with the Control and Dynamical Systems, California

Institute of Technology, Pasadena, CA 91125, USA {mgomez,syou,murray}@caltech.edu

findings highlighting the positive impact of delays on systemdynamics in biological systems. Authors Longo et al in [6]and Bhartiya et al in [7] demonstrate robust behavior ofgenetic regulatory networks with multiple delayed feedbacks.

In some biological systems, feedback loops exhibit atime delay through the involved chemical reactions thattake time to synthesize the proteins that regulate or activategene expression. Furthermore, additional delays can be arti-ficially implemented in transcription and translation throughplacement of the gene with respect to the promotor regionand secondary structure design respectively. Therefore, inorder to avoid building complex synthetic networks, thispaper aims to explore the possibility of implementing simpleproportional feedback where the time delayed output signalis the only component used in the controller design. Multipledelayed feedback channels are added to improve performanceof the system.

Previous work [8,9] shows that multiple delayed feedbackchannels can stabilize a system and [10] concentrates on theanalysis of systems with stochastic delays whose mean dy-namics resemble a distributed delay system. To some extent,we now address a synthesis problem for these biologicalsystems with mean dynamics, where we design a optimaldistribution function for stochastic delays.

As an example, a scalar genetic regulatory network isconsidered, which is described by a nonlinear dynamicalsystem. Ideally, feedback controller design should be donein the continuous-time domain with the nonlinear dynamics.However even stability analysis of a linear system withtime delays in the continuous-time domain remains a hardproblem. Therefore we consider a multiple delayed feedbackdesign for a discrete-time linear scalar system. To applyour method to a nonlinear dynamical system, we first lin-earize the system, and convert the continuous-time systemto a discrete-time system by periodic sampling. We thenimplement this controller in the continuous-time domain withnonlinear dynamics to check the closed loop stability andperformance.

This paper is organized as follows. Section II presents themultiple delayed feedback design problem for a discrete-time scalar linear system. Section III introduces a conecomplementarity linearization algorithm to get a suboptimalsolution of the design problem. Section IV describes themethod to convert a continuous-time system to a discretetime system, Section V contains an application to a scalargenetic regulatory network, and Section VI concludes thepaper.

Submitted, 2014 American Control Conference (ACC)http://www.cds.caltech.edu/~murray/papers/gym14-acc.html

II. PROBLEM FORMULATION

Consider a scalar discrete-time linear system with inherentdelay ⌧ � 1,

xk+1 = a x

k

+ b xk�⌧

. (1)

We would like to improve performance and robustness of thesystem by adding additional delayed feedbacks. The systembecomes

xk+1 = a x

k

+ b

NX

i=⌧

wi

xk�i

, (2)

where ⌧ and N � ⌧ are positive integers. In addition, werequire

PN

i=⌧

wi

= 1, and wi

� 0. Although, one can alsoconsider negative gains, we only consider positive gains inorder to demonstrate the effect of the delay distribution alone.Constraining the sum of the weights to be unity ensuresno change in the overall effective feedback gain. Later,we demonstrate the implementation of such a controller ina genetic regulatory network. Although, negative weightscan also be considered in the synthesis of these networks,they require a different set up since it involves a differentnonlinearity.

Consider an input disturbance ⌘k

such that

xk+1 = a x

k

+ b

NX

i=⌧

wi

xk�i

+ ⌘k

.

Suppose we want to minimize the effect from the inputdisturbance to the state. The transfer function from the inputdisturbance to the state is given by

H⌘!x

(z) =P

1 + PC, (3)

where P and C are the z-transforms of the plant andcontroller:

P =1

z � a

C = �b

NX

i=⌧

wi

z�i.

This results in the following transfer function,

H⌘!x

(z) =1

z � a� bP

N

i=⌧

wi

z�i

. (4)

Assuming unwanted disturbances reside in the higher fre-quency regime, we apply a weighting function [11], namely,a pre-determined filter F to ensure signals at higher frequen-cies are more greatly penalized. This is achieved with a firstorder high-pass filer

F (z) =(1 + ↵)z

z � ↵,

where 0 ↵ < 1, resulting in the following problem:

minimizew

⌧

,··· ,wN

||FH⌘!x

||1

subject toNX

i=⌧

wi

= 1, wi

� 0.(5)

The objective function is the H1 norm of the transferfunction,

||FH⌘!x

||1 = sup||⌘||21

||x||2||⌘||2

,

which is the worst-case 2 norm of the state induced bythe input disturbance with unit 2 norm. Since F is a highpass filter, the high frequency gain in the transfer functionis attenuated by solving the optimization (5). Using thegeneralized plant model, we can convert the optimization(5) to a static output feedback H1 problem with additionalaffine constraints on the gain. The state dynamics can bere-written as

x

k+1 = A0 xk

+B1 ⌘k +B2 uk

zk

= C1 xk

+D11 ⌘k +D12 uk

y

k

= C2 xk

uk

= wy

k

, (6)

where

A0 =

2

66666664

a 0 · · · 0 0 01 0 · · · 0 0 00 1 · · · 0 0 0...

.... . .

......

...0 0 · · · 1 0 0↵ 0 · · · 0 0 ↵

3

77777775

,

B1 =⇥1 0 · · · 0 0

⇤T

,

B2 =⇥b 0 · · · 0 0

⇤T

,

C1 =⇥1 + ↵ · · · 0 1 + ↵

⇤,

D11 = 0, D12 = 0,

C2 =⇥0

d⇥⌧

I

d⇥d

0

d⇥1

⇤,

w =⇥w

⌧

w⌧+1 · · · w

N

⇤,

and d = N � ⌧ + 1. In addition, the state vector x

k+1 =⇥xk+1 x

k

· · · xk�N+1

⇤T 2 RN+1, u 2 R and y 2

RN�⌧+1. The optimization problem is reformulated as

minimizew

⌧

,··· ,wN

sup||⌘||21

||x||2||⌘||2

subject to 1

T

w = 1,wi

� 0.

(7)

We refer to the gain w as the delay distribution. The staticoutput feedback H1 design is well studied in the literature,see [12, 13]. This problem is known to be non-convex, andhard in some cases [14], however, in practice there exists agood solver which gives a reasonable solution in many cases.

The discrete-time KYP lemma [15] converts the H1 normminimization problem (7) into the following optimizationproblem with Linear Matrix Inequalities (LMI):

minimizew,X,Y,�

�

subject to 1

T

w = 1,w ⌫ 0,

XY = I,X � 0,

T � 0,

(8)

where

T =

2

664

�Y A0 +B2wC2 B1 0

⇤ �X 0 (C1 +D12wC2)T

⇤ ⇤ �� D

T

11

⇤ ⇤ ⇤ ��

3

775 .

For a review of LMI and convex optimization, see [16].Notice that this problem is non-convex because of the non-affine equality XY = I. In some cases, one can convertthe non-convex problem (8) to a convex one through achange of variables [17]. However, (8) does not satisfy thenecessary assumptions in [17]. Therefore, we use a conecomplementarity linearization algorithm [18] to handle thebilinear matrix equality.

III. A CONE COMPLEMENTARITY LINEARIZATIONALGORITHM

Since the optimization (8) is not convex, we present anapproximate semidefinite program which gives us a subopti-mal solution of the problem (8). Let us start with followingobservation.

Proposition 1: The optimal solution of following prob-lem, (X?,Y?), satisfies X

?

Y

? = I.

minimizeX,Y

Tr(XY)

subject to

X I

I Y

�⌫ 0

X � 0

Y � 0.

(9)

Proof: By taking Schur complement to the first LMI,we have, Y�X

�1 ⌫ 0. This is equivalent to X

1/2YX

1/2 ⌫I. Since Tr(X1/2

YX

1/2) = Tr(XY), the minimum isachieved when X

1/2YX

1/2 = I. This implies XY = I.

Now for a given �, consider the following optimizationproblem:

minimizew,X,Y

Tr(XY)

subject to 1

T

w = 1,w ⌫ 0,

X,Y ⌫ 0,

X I

I Y

�⌫ 0 T � 0,

where

T =

2

664

�Y A0 +B2wC2 B1 0

⇤ �X 0 (C1 +D12wC2)T

⇤ ⇤ �� D

T

11

⇤ ⇤ ⇤ ��

3

775 .

If there exists a triplet (w,X,Y) that satisfies all theconstraints and XY = I, then the above optimizationproblem recovers this triplet. Therefore, we can successfullyconstruct a w such that the H1 norm of the transfer functionis less than �. However, since the objective function is

not convex, we use a linearization technique to solve thisproblem, namely:

minimizew,X,Y

Tr(Xk

Y +XY

k

)

subject to 1

T

w = 1,w ⌫ 0,

X,Y ⌫ 0,

X I

I Y

�⌫ 0

T � 0.

(10)

Cone complementary solver:

1) Set k = 0, and X0 = Y0 = I .2) Solve the optimization problem (10) to generate X

k+1,Y

k+1.3) Set k = k + 1, and do step 2 until X

k

converges.Note that if iterative procedure above finds (w,X,Y) whereXY = I, then this weight w guarantees the stability ofthe closed loop system and the H1 norm is less than �.However, since the above procedure uses a linearized versionof the true objective function, the procedure may fail torecover the solution (w,X,X�1) in some cases. In thissense, this procedure is only an approximate solver for theoriginal problem. In practice, this approach works well.

Finally, since we can approximately solve the problemgiven �, we now apply a bisection search to obtain theminimum �.

Bisection search:

1) Set l = 0, and � = 1.2) Solve the optimization problem (10) for �.3) If (10) recovers XY = I, then set u = �, otherwise

l = �.4) Set � = 1

2 (l + u). Repeat step 2 until � converges.Again, since (10) is an approximate solver, the above bi-section search can converge to a point which is not a trueminimum.

IV. CONTINUOUS TIME CONVERSION

We briefly summarize a method used to convert acontinuous-time linear system to a discrete-time linear sys-tem.

Suppose we have a continuous-time linear dynamicalsystem,

dx

dt= Ax+B u.

Consider we sample x(t) with sampling period T , then, thissystem generates sequence x

k

= x(kT ). The zero order hold(ZOH) method [19] assumes u(t) remains constant duringthe each sampling period [kT, (k + 1)T ]. Assuming u

k

=u(kT ), we can convert the above continuous-time system tothe following discrete time system,

x

k+1 = eAT

x

k

+A�1(eAT � I)Bu

k

.

However, biological systems, which are a main targetapplication, are usually described by nonlinear dynamicalsystems. Henceforth, we linearize the nonlinear dynamics

and then apply the ZOH method to the linearized system.From the obtained delay distribution vector w, we appropri-ately incorporate the distributed delayed feedback into theoriginal nonlinear dynamics. The full nonlinear dynamics aresimulated to check performance and stability.

To draw a bode plot in s-domain from the data in thez-domain, we use the following method. Suppose one hasa transfer function in the z-domain, H(z), then substitutez = ejwT to obtain a complex valued function H(ejwT ). Plot|H(ejwT )| as a Bode magnitude plot by varying w 2 [0, ⇡

T

].For details, see [20].

V. APPLICATION TO A SCALAR GENETIC REGULATORYNETWORK

A. Basic Model

Fig. 1. Self regulating protein production.

In this section we apply the numerical method to thedesign of a scalar auto-regulatory genetic system. We take ascalar example from [21],

x = �ptot

x/Kd

+ 1�� x+

s� p

tot

Ae

/Kd

+ 1n1�

p� xn2, (11)

where x represents the concentration of a self regulatingprotein. The rates � and � are the production and degradationrates respectively. In addition, n1 and n2 represent the noiseson production and decay respectively. An example of thesystem is shown in Fig. 1, where protein x inhibits furtherproduction of itself by not allowing the RNA polymerase tobind and, therefore, inhibiting initiation of transcription. Theconstants p

tot

and Kd

represent the total concentration of theDNA promotor and the ratio of dissociation to associationrates.

We will consider only noise on protein production (n2 =0). We modify the model to account for delays due totranscription and translation,

x = �ptot

x(t� ⌧)/Kd

+ 1� � x+

s� p

tot

Ae

/Kd

+ 1n1.

The system is then linearized around the equilibrium pointxe

= 12

⇣pK2

d

+ 4� ptot

Kd

/� �Kd

⌘by treating the delay

channel as an input which gives

x = �� x(t) + x(t� ⌧) + b1 n1,

where = �� p

tot

/K

d

(xe

/Kd+1)2 , b1 =q

� p

tot

x

e

/K

d

+1 and b2 =

�p� x

e

. Applying the zero order hold method, an equivalentdiscrete time system is given by,

x(k + 1) = d2 x(k) + d1 x(k � ⌧) + d1 b1 n1(k)

where d2 = e�T� and d1 = 1�

(1 � e�T�). By assumingthat the delay is a multiple of the sampling time T , wecan arrive at a discrete-time system with multiple delayedfeedback channels:

x(k + 1) = d2 x(k) + d1

NX

i=⌧

wi

x(k � i) + d1 b1 n1(k).

We optimize over the weights and compare to the originalsystem. The system parameters are partially taken from [21]and chosen to be p

tot

= 10, Kd

= 10, � = .1, and � = .01.

B. Design Results

Bode Magnitude Plot for different filters

Frequency (Hz)10

!510

!410

!310

!2!30

!25

!20

!15

!10

!5

0M

agnit

ude

(dB

)

0.10.20.30.40.50.60.70.80.9

Fig. 2. First order highpass filter, (1+↵)zz�↵ , with multiple ↵.

We first present various high pass filters, (1+↵)zz�↵

, byvarying ↵ in Fig. 2. Note, a larger ↵ results in a smallercut-off frequency. Therefore, depending on our assumptionon the noise, we can choose ↵ to obtain an appropriatesensitivity transfer function.

For the discretization, we use T = 60s (1 minute) as asampling period, and assume the minimum time delay ⌧ =10, which corresponds 10 minutes. For maximum delays, wetest N = 10, 15, 20, 25 and 30 as an example, and set thefilter parameter ↵ = 0.99.

Fig. 3 shows the bode magnitude pot of the transferfunction for various maximum delays. A larger maximumdelay N results in more degrees of freedom in the designstage, hence, we should expect better performance in theresult. As we can see, when N = 30 the transfer functionH = P

1+PC

shows better attenuation. Notice that we opti-mized the filtered version of the H1 norm, ||FH||1 and

Bode Magnitude Plot for Different Maximum Delays

Frequency (Hz)10

!510

!410

!310

!2!10

!5

0

5

10

15

20

Mag

nit

ud

e (d

B)

N=10N=15N=20N=25N=30

Fig. 3. Resulting transfer function P1+PC for multiple maximum delays

N . The lower, the better.

10 15 20 25 300

0.2

0.4

0.6

0.8

1

Delay

Wei

ght

Delay Distribution

N=10N=15N=20N=25N=30

Fig. 4. Resulting delay distribution for multiple maximum delays N .

not the unweighted one, so a direct numerical comparisonof Bode magnitude plots should be considered carefully.

For the corresponding delay distribution vector see Fig. 4.Notice that the first and the last delay channels are mostimportant, and other channels have similar weights to eachother.

C. Time Simulation

We apply the results obtained to the original nonlinearequation (11) with n2 = 0. For a general distributed delayrepresentation, (11) can be re-written as

x = �ptot

x/Kd

+ 1� � x+

s� p

tot

Ae

/Kd

+ 1n1

x =

Z 1

0x(t� ⌧)g(⌧)d⌧

where g(t) is the delay distribution function such thatR10 g(⌧)d⌧ = 1, and g(t) � 0 for all t. In the discretization

of the system, g(t) takes on the form

g(t) =

NX

k=⌧

wk

�(⌧ � kT )

whereP

N

k=⌧

wi

= 1, T is the sampling period, and � isthe Dirac delta function. The final model representing themodified network is described by

x(t) = �ptot

(P

N

k=⌧

w

k

x(t�kT ))K

d

+ 1�� x(t)+

s� p

tot

Ae

/Kd

+ 1n1.

One way the delays can be implemented is by adding ”junk”DNA in between the promotor site and the gene to delaythe initiation of transcription. The ”junk” DNA would be asequence of DNA strand that does not code for anything,but that the RNA polymerase would still need to transcribebefore beginning transcription of the target gene. Since theoverall gain remains unchanged there is no need to changeany of the parameters in the system such as the productionrate. This, in general, may be harder to do. However, onedoes need to consider how to account for the differencesin the weights. This can possibly be implemented throughproportions of various plasmids with the different delaysimplemented so that the ratios on concentrations correspondto the weights. A more challenging but possibly usefulapproach would be to apply competitive binding so thatthe weights correspond to the probability of each particulardelayed state binding to the promotor site.

Before presenting simulations, the stability regions forthe different sets of delays and their respective weights areconsidered. This is important to take note of when consid-ering how the system stability may change with respect touncertainty in the parameters of the system when addingdelays. The stability region for the general system (2) isdetermined by the characteristic equation

z � a� b

NX

i=⌧

wi

z�i = 0

given by the denominator of the transfer function (4). Fordiscrete-time systems, a system is unstable when |z| � 1,therefore, the characteristic equation is evaluated at |z| = 1to map out the curves on the (a, b) parameter space whenan eigenvalue crosses the unit circle. These curves can beobtained by evaluating the characteristic equation at z = 1,z = �1, and z = ei✓ [22,23]. It is worth noting z = 1(black curve) gives a delay independent stability conditionb = 1� a. However, z = �1 (red curve) and z = ei✓ (bluecurves) give

b =�(a+ 1)

PN

i=⌧

wi

(�1)i

and

b(✓) = � sin(✓)/

NX

i=⌧

wi

sin(i✓)

!

a(✓, b) = cos(✓)� b

NX

i=⌧

wi

cos(i✓)

!

respectively.

!1.5 !1 !0.5 0 0.5 1 1.5!1.5

!1

!0.5

0

0.5

1

1.5

a

b

N=10

Fig. 5. Stability plot for the system with a single delay ⌧ = 10. The blackcircle indicates the parameter values corresponding to the simulations.

Fig. 5 shows the stability region for the original systemwith a single delay. If the system parameters (a, b) are in theshaded region, then the system is stable. From our nominalparameter values, we obtain (a, b) = (0.5488,�0.3293),which is indicated by a circle marker in Fig. 5.

Fig. 6 shows the stability regions for the various distributeddelayed systems. The overlaying green shaded region andgrey curves correspond to the stability region for the originalsystem shown in Fig. 5. One can see that the area of thestability region increases as the maximum delay N increases.There is also a notable difference in the robustness touncertainty in parameter values.

Lastly, the nonlinear system with multiple delayed feed-back is simulated in Fig. 7. The noise is modeled as a sumof periodic functions at various equally spaced frequenciesbeginning at 10�3Hz. Notice that in Fig. 7, our designedmultiple delayed feedback channels help to decrease thefluctuations in protein production.

VI. CONCLUSION

Inspired by biological applications, we consider the sub-optimal multiple delayed feedback channel design for ascalar linear discrete-time system. We formulate a H1

!1.5 !1 !0.5 0 0.5 1 1.5!1.5

!1

!0.5

0

0.5

1

1.5

a

b

N=15

!1.5 !1 !0.5 0 0.5 1 1.5!1.5

!1

!0.5

0

0.5

1

1.5

a

b

N=20

!1.5 !1 !0.5 0 0.5 1 1.5!1.5

!1

!0.5

0

0.5

1

1.5

a

b

N=25

!1.5 !1 !0.5 0 0.5 1 1.5!1.5

!1

!0.5

0

0.5

1

1.5

a

b

N=30

Fig. 6. Stability plots for different delay ranges and distributions (blue).The shaded green region and grey lines correspond to the stability plot inFig. 5. The black circle indicates the parameter values corresponding to thesimulations.

0 20 40 60 80 100 120 140 16015

20

25

30

35

40

time [min]

x(t

)

N=10N=15N=20N=25N=30

Fig. 7. Simulations of nonlinear system with distributed delays.

design problem with multiple delayed feedback, and make aconnection to a static output feedback H1 with additionallinear constraints on a static gain. We apply this designprocedure to a sampled, linearized scalar auto-regulatorygenetic system, and show that the effect from the inputdisturbance can be suppressed by adding appropriate multipledelays. Furthermore, we simulate an auto-regulatory geneticsystem with multiple delays, and show, indeed, the fullnonlinear system has improved performance in the presenceof stochastic protein production.

Notice, that even though adding additional delays de-creases the H1 norm from the input to the output, thewaterbed affect can result in larger magnitudes at the higherfrequencies. It appears as though the distribution which givesthe smallest H1 norm may not always be the best choice.One could imagine designing the weighting filter such thatimprovement is obtained where needed, based on knowledgeof uncertainty.

In the future, we generalize this concept to a multi-statesystem. We would also like to explore removing the positivedefinite constraint on the weights in order to allow for designwith a combination of positive and negative feedback loops.Another potential application to explore is that of achievingplant stability in a car following model with delays. Plantstability is achieved when the H1 norm of the transferfunctions from every possible source of input to outputremain equal to or below unity. This method may provide asimpler method of improving performance with minimizedcomputational cost.

VII. ACKNOWLEDGMENTS

The authors gratefully acknowledge the helpful discus-sions regarding genetic regulatory networks and biologicalprocess with the Murray lab, as well as Erin O’Brien andAndrew Hirning from the Bennett lab at Rice University.This work was supported in part by the TerraSwarm ResearchCenter, one of six centers supported by the STARnet phaseof the Focus Center Research Program (FCRP) a Semicon-ductor Research Corporation program sponsored by MARCOand DARPA, and the Kwanjeong Graduate Fellowship.

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